Abstract
A coordinate is an element of a polynomial ring which is the first component of some automorphism of this ring. Understanding the structure of coordinates is still one of the major problems in the theory of polynomial automorphisms. It is already known, that in two variables over a field every coordinate is tame. The structure of coordinates over an arbitrary commutative ring, however, is still very unclear, and therefore worth investigating, also because many well-known non-tame coordinates in more than two variables emanate from them. This thesis applies the 'stabilization'-concept (i.e., viewing a problem in a higher dimension) to polynomials over a commutative ring. It is shown that over a Noetherian domain with Krull dimension one, every coordinate is stably tame. Much attention is also paid to the structure of a special class of polynomials in two variables over an arbitrary commutative ring. Their characteristics are very crucial to the main theory. But we also focus on (stable) tameness of automorphisms in two variables. Leading coefficients of coordinates play a central role here. A special chapter is devoted to a comprehensive algorithm to recognize coordinates in two variables over a finitely generated algebra over a computable field of characteristic zero. Also, in case of a coordinate we give an algorithm to compute a mate. Lastly, we present a few techniques to construct new coordinates out of given ones.
